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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bayesian multivariate linear regression ： ウィキペディア英語版 Bayesian multivariate linear regression In statistics, Bayesian multivariate linear regression is a Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator. ==Details== Consider a regression problem where the dependent variable to be predicted is not a single real-valued scalar but an ''m''-length vector of correlated real numbers. As in the standard regression setup, there are ''n'' observations, where each observation ''i'' consists of ''k''-1 explanatory variables, grouped into a vector $\backslash mathbf\_i$ of length ''k'' (where a dummy variable with a value of 1 has been added to allow for an intercept coefficient). This can be viewed as a set of ''m'' related regression problems for each observation ''i'': :$y\_\; =\; \backslash mathbf\_i^\backslash boldsymbol\backslash beta\_\; +\; \backslash epsilon\_$ :$\backslash cdots$ :$y\_\; =\; \backslash mathbf\_i^\backslash boldsymbol\backslash beta\_\; +\; \backslash epsilon\_$ where the set of errors $\backslash \backslash \}$ are all correlated. Equivalently, it can be viewed as a single regression problem where the outcome is a row vector $\backslash mathbf\_i^$ and the regression coefficient vectors are stacked next to each other, as follows: :$\backslash mathbf\_i^\; =\; \backslash mathbf\_i^\backslash mathbf\; +\; \backslash boldsymbol\backslash epsilon\_^.$ The coefficient matrix B is a $k\; \backslash times\; m$ matrix where the coefficient vectors $\backslash boldsymbol\backslash beta\_1,\backslash ldots,\backslash boldsymbol\backslash beta\_m$ for each regression problem are stacked horizontally: :$\backslash mathbf\; =$ \begin \begin \\ \boldsymbol\beta_1 \\ \\ \end \cdots \begin \\ \boldsymbol\beta_m \\ \\ \end \end = \begin \begin \beta_ \\ \vdots \\ \beta_ \\ \end \cdots \begin \beta_ \\ \vdots \\ \beta_ \\ \end \end . The noise vector $\backslash boldsymbol\backslash epsilon\_$ for each observation ''i'' is jointly normal, so that the outcomes for a given observation are correlated: :$\backslash boldsymbol\backslash epsilon\_i\; \backslash sim\; N(0,\; \backslash boldsymbol\backslash Sigma\_^2).$ We can write the entire regression problem in matrix form as: :$\backslash mathbf\; =\backslash mathbf\backslash mathbf\; +\; \backslash mathbf,$ where Y and E are $n\; \backslash times\; m$ matrices. The design matrix X is an $n\; \backslash times\; k$ matrix with the observations stacked vertically, as in the standard linear regression setup: :$$ \mathbf = \begin \mathbf^_1 \\ \mathbf^_2 \\ \vdots \\ \mathbf^_n \end = \begin x_ & \cdots & x_ \\ x_ & \cdots & x_ \\ \vdots & \ddots & \vdots \\ x_ & \cdots & x_ \end. The classical, frequentists linear least squares solution is to simply estimate the matrix of regression coefficients $\backslash hat\}\; =\; (\backslash mathbf^\backslash mathbf)^\backslash mathbf^\backslash mathbf$. To obtain the Bayesian solution, we need to specify the conditional likelihood and then find the appropriate conjugate prior. As with the univariate case of linear Bayesian regression, we will find that we can specify a natural conditional conjugate prior (which is scale dependent). Let us write our conditional likelihood as〔Peter E. Rossi, Greg M. Allenby, Rob McCulloch. ''Bayesian Statistics and Marketing''. John Wiley & Sons, 2012, p. 32.〕 :$\backslash rho(\backslash mathbf|\backslash boldsymbol\backslash Sigma\_)\; \backslash propto\; (\backslash boldsymbol\backslash Sigma\_^)^\; \backslash exp(-\backslash frac\; (\backslash mathbf^\; \backslash mathbf\; \backslash boldsymbol\backslash Sigma\_^)\; )\; ,$ writing the error $\backslash mathbf$ in terms of $\backslash mathbf,\backslash mathbf,$ and $\backslash mathbf$ yields :$\backslash rho(\backslash mathbf|\backslash mathbf,\backslash mathbf,\backslash boldsymbol\backslash Sigma\_)\; \backslash propto\; (\backslash boldsymbol\backslash Sigma\_^)^\; \backslash exp(-\backslash frac\; ((\backslash mathbf-\backslash mathbf\backslash mathbf\; (\backslash mathbf-\backslash mathbf\backslash mathbf^\; )\; )\; ,$ We seek a natural conjugate prior—a joint density $\backslash rho(\backslash mathbf,\backslash Sigma\_)$ which is of the same functional form as the likelihood. Since the likelihood is quadratic in $\backslash mathbf$, we re-write the likelihood so it is normal in $(\backslash mathbf-\backslash hat|\backslash mathbf,\backslash mathbf,\backslash boldsymbol\backslash Sigma\_)\; \backslash propto\; \backslash boldsymbol\backslash Sigma\_^\; \backslash exp(-(\backslash frac\backslash mathbf^\backslash mathbf\; \backslash boldsymbol\backslash Sigma\_^))$ (\boldsymbol\Sigma_^)^ \exp(-\frac ((\mathbf-\hat \mathbf^\mathbf(\mathbf-\hat^ ) ) , :$\backslash mathbf\; =\; \backslash mathbf\; -\; \backslash hat$ We would like to develop a conditional form for the priors: :$\backslash rho(\backslash mathbf,\backslash boldsymbol\backslash Sigma\_)\; =\; \backslash rho(\backslash boldsymbol\backslash Sigma\_)\backslash rho(\backslash mathbf|\backslash boldsymbol\backslash Sigma\_),$ where $\backslash rho(\backslash boldsymbol\backslash Sigma\_)$ is an inverse-Wishart distribution and $\backslash rho(\backslash mathbf|\backslash boldsymbol\backslash Sigma\_)$ is some form of normal distribution in the matrix $\backslash mathbf$. This is accomplished using the vectorization transformation, which converts the likelihood from a function of the matrices $\backslash mathbf,\; \backslash hat(\backslash mathbf),\; \backslash hat\; =\; (\backslash hat((\backslash mathbf\; -\; \backslash hat\backslash mathbf^\; \backslash mathbf(\backslash mathbf\; -\; \backslash hat^)\; =\; (\backslash mathbf\; -\; \backslash hat(\backslash mathbf^\; \backslash mathbf(\backslash mathbf\; -\; \backslash hat^\; )$ Let :$(\backslash mathbf^\; \backslash mathbf(\backslash mathbf\; -\; \backslash hat^\; )\; =\; (\backslash boldsymbol\backslash Sigma\_^\; \backslash otimes\; \backslash mathbf^\backslash mathbf\; )(\backslash mathbf\; -\; \backslash hat\; \backslash otimes\; \backslash mathbf$ denotes the Kronecker product of matrices A and B, a generalization of the outer product which multiplies an $m\; \backslash times\; n$ matrix by a $p\; \backslash times\; q$ matrix to generate an $mp\; \backslash times\; nq$ matrix, consisting of every combination of products of elements from the two matrices. Then :$(\backslash mathbf\; -\; \backslash hat\; (\backslash boldsymbol\backslash Sigma\_^\; \backslash otimes\; \backslash mathbf^\backslash mathbf\; )(\backslash mathbf\; -\; \backslash hat)^(\backslash boldsymbol\backslash Sigma\_^\; \backslash otimes\; \backslash mathbf^\backslash mathbf\; )(\backslash boldsymbol\backslash beta-\backslash hat)$ which will lead to a likelihood which is normal in $(\backslash boldsymbol\backslash beta\; -\; \backslash hat)$. With the likelihood in a more tractable form, we can now find a natural (conditional) conjugate prior. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bayesian multivariate linear regression」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.